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Mirrors > Home > MPE Home > Th. List > mndidcl | Structured version Visualization version GIF version |
Description: The identity element of a monoid belongs to the monoid. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.) |
Ref | Expression |
---|---|
mndidcl.b | ⊢ 𝐵 = (Base‘𝐺) |
mndidcl.o | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
mndidcl | ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mndidcl.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | mndidcl.o | . 2 ⊢ 0 = (0g‘𝐺) | |
3 | eqid 2651 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | 1, 3 | mndid 17350 | . 2 ⊢ (𝐺 ∈ Mnd → ∃𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(+g‘𝐺)𝑦) = 𝑦 ∧ (𝑦(+g‘𝐺)𝑥) = 𝑦)) |
5 | 1, 2, 3, 4 | mgmidcl 17312 | 1 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 Basecbs 15904 +gcplusg 15988 0gc0g 16147 Mndcmnd 17341 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-iota 5889 df-fun 5928 df-fv 5934 df-riota 6651 df-ov 6693 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 |
This theorem is referenced by: mndpfo 17361 prdsidlem 17369 imasmnd 17375 idmhm 17391 mhmf1o 17392 issubmd 17396 submid 17398 0mhm 17405 mhmco 17409 mhmeql 17411 submacs 17412 mrcmndind 17413 prdspjmhm 17414 pwsdiagmhm 17416 pwsco1mhm 17417 pwsco2mhm 17418 gsumvallem2 17419 dfgrp2 17494 grpidcl 17497 mhmid 17583 mhmmnd 17584 mulgnn0cl 17605 mulgnn0z 17614 cntzsubm 17814 oppgmnd 17830 gex1 18052 mulgnn0di 18277 mulgmhm 18279 subcmn 18288 gsumval3 18354 gsumzcl2 18357 gsumzaddlem 18367 gsumzsplit 18373 gsumzmhm 18383 gsummpt1n0 18410 srgidcl 18564 srg0cl 18565 ringidcl 18614 gsummgp0 18654 pwssplit1 19107 dsmm0cl 20132 dsmmacl 20133 mndvlid 20247 mndvrid 20248 mdet0 20460 mndifsplit 20490 gsummatr01lem3 20511 pmatcollpw3fi1lem1 20639 tmdmulg 21943 tmdgsum 21946 tsms0 21992 tsmssplit 22002 tsmsxp 22005 submomnd 29838 omndmul2 29840 omndmul3 29841 omndmul 29842 ogrpinv0le 29844 slmdbn0 29889 slmdsn0 29892 slmd0vcl 29902 gsumle 29907 sibf0 30524 sitmcl 30541 pwssplit4 37976 c0mgm 42234 c0mhm 42235 c0snmgmhm 42239 c0snmhm 42240 mgpsumz 42466 mndpsuppss 42477 lco0 42541 |
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